Left-Right. Symmetry and Neutrino Masses in a. Non-Perturbative. Unification. Framework. P.Q. Hung and S. Mohan'. Fermi National Accelerator Laboratory.
FERMILAB-PUB-87/212-T November,
Left-Right Symmetry Non-Perturbative
P.Q. Hung Fermi National
and Neutrino Masses Unification Framework
1987
in a
and S. Mohan’ Accelerator Laboratory
P. 0. Boz 500, Batavia, IL 60510
Abstract Within
the non-perturbative
unification
framework pioneered by Maiani,
Parisi and Petronzio, it is shown how the mass of the right-handed W boson can be determined just by knowing the values of sin’ 8, and ag(= gz/4lr) at the Fermi scale Ap u 250 GeV. Consequently, the knowledge of A4wR helps to determine the light Majorana neutrino mass which is my 2 O(l/Mw,). The best bounds we can obtain here are Mw, X 8 TeV, m,, s 5 x 10~eel’, rn,* S 0.2eV, rn”, s 65ev. lP.Q. Hung and S. Mohan on leave of absencefrom Department of Phyaies, University of Virginia, Charlottesville, VA 22901
-l-
The mystery
of parity
merous investigations its simplest
violation
Unified
can only be understood
extension,
example
broken
parity
symmetric
hand,
This idea is quite attractive
modelsl’l
and it would
on the value of AR. The Sum
breaking
to be AF = (&Gr)-i
for Aa except for the lower limits
has sparked nu-
model as well as in
violation
of the standard it is quite
at some scale, An, above which
of left-right
experimentally
i.e., SU(5),
On the other
interactions
In the standard
origin.
if one knows the origin
and its gauge interactions. spontaneously
in the electrowesk
into its possible
Grand
FERMILAB-Pub-871212-T
is intrinsic particle
possible
spectrum
that
it is restored.
of the type
sum
be very helpful
parity
This
SU(2)n
x
and is
is the x
U(1).
if one could get a handle
scale or the Fermi scale AF is determined
c1 250 GeV. No such determination on the mass of the SU(2)a
is found
gauge bosons, namely
MwR > 300 GeV or 1 TeV. Is it somehow possible to relate the two scales AF and An? This is the question would like to address in this letter.
It turns out that under a certain
assumptions,
does exist.
such a relationship
to calculate
sin20,(AF)
where An is with implications
and as(A,)
on neutrino
in this note was advanced by Maiani,
dard model131 are the near-infrared It means two things: U(l),)
x
are small;
asymptotically
free (U(l),
at which
perturbation
point
singularity).
that the “low”
At AF, as,
theory
couplings
sin* ~,(A.w) and as(Ap)
of standard
extra families no Grand
ingredient and os(A~)
of the stanfree
(of SU(3),
are themselves
non-
(this is the famous
Landau
to how large
enabled the authors
of Refs.
(Ref. [5] also applied
the same method
in the MPP scenario
is a relatively
In general, this number
to have reasonable Another
x
grow large at some scale Ao to be insensitive
are. These considerations
were given mass of O(4).
Unified
are found
quark and lepton families.
and 10 in order for sin’ e,(A,)
and SU(2)t
ceases to be valid
[2,4,5] to compute
A crucial
and ov(A.~)
is) and all couplings
couplings
sector).
has deep
of a non-asymptotically
or,
above AF, SU(3),
the “high”-energy
number
determine
Parisi, and Petronsio[21
energy couplings
stable fixed points
already
The “lawn-energy
the Yukawa
us
masses.
several years ago. It is the assumption
sum
in turn,
allow
respect to AF. It also turns out that this determination
The concept employed
theory.
set of reasonable
In fact, these assumptions
and such computations,
we
crucial
values.
element
gauge group was assumed in the computation
to
large
is between
8
Many of the
is the fact that
of sin2 E’,(A,).
FERMILAB-Pub-871212-T
-2-
The simplest
extension
of the standard
x SU(~)J, x SU(2)R
x U(l)‘.
model with,
the calculation
in mind,
the right-handed
breaking
The following
We now apply
set of fermion
of sin’ fl,(A,),
we have $i
= (3,2,1,
numbers.
right-handed
and the connection
2, -1)
N o t ice that,
B and L are the baryon
The set $L,R Iq forms a standard
family
Ref. [l].
of
For
while for the quarks
U(l)’
here, we identify
and lepton
the last entries
Since in this case Q = 2’s~ + 2’s~ + v, quantum
os(A~),
and I& = (l,l,
-1)
5) and $k = (3,1,2,5).
where
scheme to this extended
and scalar fields are chosen following
we have 4: = (1,2,1,
U(l),-,,
the MPP
is SU(3),
symmetry
scale AR to the Fermi scale AF.
the leptons, with
model with left-right
numbers
respectively.
denote the B - L
in tip,
where one now also has a
neutrino.
The Higgs fields include
the following
sets:
1. ~=(1,2,2,0),A~=(1,2,1,1),A~=(1,1,2,1)or 2. ~$=(1,2,2,0),A~=(1,3,1,2),A~=(l,1,3,2). The first set (4, AL, Aa) parable
gives only Dirac masses to the neutrinos
to those of the charged
the neutrino
leptons.
masses, one has to put Majorana
see-saw mechanism gauge invariance
to obtain
gives Dirac masses to the neutrinos masses through
this point
later in this paper.
seen that
This process, however,
As pointed through
the couplings
with
out by Ref.
the coupling
of
violates
of symmetry
breaking
[l], this set not only
with
AL and An.
the above set of Higgs fields whose number the pattern
the smallness
even if we had MV~VR with M 5 AR. The second
and natural.
Majorana
are com-
mass terms in by hand and use the
the mass eigenvalues.
and is unnatural
set is more interesting
With
In this case, to explain
which
4, but it also gives
We will
come back to
is left arbitrary,
is as follows:
it can be
SU(2)& x sum
x
The one-loop renormalization group U(l)&L -‘Aa SU(2)L x U(l), -+hF U(l).,. (RG) equations describing the evolution of the couplings above An are given by dQL,.Q _ T, *dt -- g1 &rI and, for AF 2 E 2 AR: by % = b2rc&., % = bya:, 2L,RQ:L,R, wtre
oi = gf/4x
Also, t = en(p2/M2). 6(A~)+n++4n),i)r~
and &i and or correspond The R.G. coefficients = I$(-22+6(A~)+n++4tZ),i;1
to U(l)s-L
and U(l),
respectively.
are given as &L = bra = A(-22
+
= &($+S(AL)+Z(AR)),by
=
-3-
&(y+f(Ar)+n+),
where n is the number
Higgs fields, 6(A)
and, in general,
= n~/2
$(A,)
when
= $(A,)
left-right
A~,R are doublets
= $5(A)
Since the theory
symmetry
n+ the number of 4(2,2,0) C-1 I-) implies 6 (An) = 6 (AR) =
and 6(An)
= 6(An)
is non-asymptotically
p-function.
Explicitly,
a, = g;/4x
with
one has %
+ 4n),Ca
and oioi
terms.
symmetry
breaking
couplings.
= &(76n
+ O(a:ar,
standard
o,yi(Ao),
What
and
model with
with
which appears in the solution
that all couplings
will be neglected
to the one-loop
so that o;r(A~)
neglect the the
as long as one does
mass much larger
follows
= 1, Z)), where
we will
approximation
means in what
to the QCD
one Higgs doublet,
scale. From here on, we will assume that assumption
is ex-
b3 and Ca are given
- 306). In what follows,
fermions
this statement
o,Tr(Ao) +b&%,
a:ai(i
The coefficients
In fact, for the standard
We now make the MPP scale Ao.
= 2%
the two loop contributions
results of Ref. (51 show that this is a reasonable not have superheavy
= 6(A)
free above AF and since as(&)
= bs(~: + C&
gf being Yukawa
by b3 = &(-33
of families,
= 3na when AL,R are triplets.
pected to be of order 0.1, one has to include
oio,
FERMILAB-Pub-871212-T
than
the SV(2)r
such is the case.
grow large at a common is simply
equation, u bitn%.
that
namely
the term cy,:‘(A~)
=
We can then readily
find sinz8,(Ap) The value of sin* e,(A,)
is given in Table
have used I..,.
= l/128.
down to U(l),
one has o;r(A~)
cz;l(An)
+ &(Fn
(-22 + S(Ar)
= (o”::))
+ 6(A,)
1 for various
Since, at AR, Su(2)n = a;i(An)
+ n+)en$
+ n4 + 4n) en”.AF Ao, n, n+ and 6(An).
x U(l),-,
+ &;‘(An).
-22 + 6 + 8 + 2n+ + yn
symmetry
values of oa which
dominates
using
$‘(A,)
=
>
fnk AF
- 67~;~. (AF)] / [--22
+6+z]>, (4
has been used, i.e.,
of AR are given in Table 1. Integrating
accommodate
is assumed to break Upon
C-1
The following
We
we obtain
-AR =ezp AC where left-right
(.l)
(-)
6 (AL)
the two-loop
= 6 (AR).
equation
Various
values
for oa, we obtain
the
are also shown in Table 1. results
emerge.
n = 8,9 standard
over the two-loop
term
The minimal families.
model with
n+ = na = 1 can only
For n 5 7, the one-loop
Cacr: for os(A~)
term
bsa$
N 0.1 and since, in this case,
FERMILAB-Pub-871212-T
-4-
b3 < O,SU(3),
is asymptotically
for n = 10, the minimal bound
and M,).
families
predicts
0.056 S os(A~)
The predictions
in the minimal
for os(A~)
model with
Ar,n
is 8 or 9. The same conclusion
(here 0.053 SZ os(A~) sin* Q,mi”(A,),
As
S 0.06 where the lower
c- 0.217 (these values of sin’ 9, are obtained
sin* @“(A,) say that,
model
to our assumption.
ma2( AR) N 0.24 and the upper to sin* 19~
corresponds
of Ml
free above AF contrary
bound
corresponds
to
from the measurements
are too small
being triplets,
and one can safely
the number
of standard
holds for the case when Ar,n
are doublets
S 0.056 for n = 10). One can lower a little
bit the value of
to say 0.21, with the effect that the upper bound on as
is altered
very little. Before we discuss the possible values of AR, it is important Ao are reasonable and os(A~).
since AR depends on Ao (Eq. (2)). The criteria
We have stated
above that
is how well does one know os(A~). finds os(M*) handle
firm
of Agoo
= &[l
+ 2?]
+ O(t-r)
mt, as result
can be computed is 0.09 ,S os(A~)
corrections
consensus
200 MeV.
there
with
of T-decay
For various mass
S
250 GeV. The
since the uncertainty
100 MeV to 300 MeV
(approximately)
and since there is also a possibility
fourth
mass
range for os(A~)
with
the following
As for sin’ e,(A,), With
model
(2) tells us that,
we will consider
in mind,
concerning
that
as a reasonable
An = Ao.
the predictions of families
As AC increases
value, An decreases. This is shown in Table 1 where the smallest
AR corresponds
to sin* B,(AF)
of the minimal
model
1.2 x 10” GeV, An CI 9
with x
10”
= 0.24. One can readily AL,R being
triplets
GeV, os(A~)
a
of
AF < AR < AC. In fact., eq.
model and for a fixed number
value of Ao for which
that
the range 0.21 S sin* Bw(AF) S 0.24.
we now first examine
AR. We require
can run from
.S 0.12.
we shall consider as reasonable
in the minimal
there is a minimum
250 GeV exists,
one 0.08 S as(A,)
the above constraints
the minimal
minimal
S
in Ag\
as, we values of
S 0.1. However,
generation
is no
is for the value
When we “run”
is flavor-dependent.
for three generations
to as, one
Measurements
= 200 f 50 MeVlsl.
9 AZ:
The question
Although
the general
in the &?S scheme to be around in which
the two-loop
where t = en&-.
on the precise value of Acon,
use the convention
will be sin’ B,(AF)
czs(A~) 5 0.06 is too small.
Including
give, on the average, the value A$ will
to see which values of
(8 or 9), from that value of
see that the best prediction
is given
by n = 8,Ao
= 0.117 and sin’ 0,(A,)
= A4r =
= 0.207. From
-5-
FERMILAB-Pub-871212-T
Table 1, one notices that when AL,R are doublets, of os(A,)
N 0.07 for sin* tJ,(AF)
for os(A~)
An = (3,1,2),
the B’s mass, left-right
the minimal
left-right
An = (1,3,2)
predicts
symmetry
implies
MuR = $gRAR N 5 x 10”
giving
discuss the neutrino can AR become?
too small a value
u 0.216 or a too small value of sin* &(AF)
u 0.117. In summary,
one 4 = (2,2,0),
one gets either
An c- 9
8 families
one has to go beyond
= gn N 1.11
MwR more, when we
GeV. We shall talk about of immediate
and
1O’l GeV. As for
x
gn = gn and, at An,g~
masses. The question
It is clear that
model with
Y 0.182
importance
is: how small
the minimal
model in order
to lower the value of An. The simplest
extension
of the minimal
more than one set of Higgs scalars.
model is the situation
We have looked at the following
na 2 2 and 2) na = 1, n+ 2 2. We first concentrate of the model in which imposed. With
The results
AL,R are triplets.
0.21 s
sin*B,(AF)
again only n = 8or9 gives acceptable 1.3
x
best results
lo’* Gel’, An = 1.4
na = 2,Ao
Again,
cases: 1) n+ =
on the more interesting
the requirement
version
AF < AR < Ao is
are shown in Table 1.
the constraints
have the following
in which there are
= An = 2
x x
S 0.24 and 0.08 .S os(A~)
results.
For case (1) with
S 0.12,
n+ = nA 2 2, we
(lowest values of AR): n = 8, n+ = nA = 2, Ao =
10’ Gel’, sin* e,(AF) 10”
GeV,sinZB,(AF)
= 0.24, aa
= 0.119; n = 9, ng =
= 0.227,as(A~)
= 0.08. For both
n = 8 and n = 9, n+ = nA 2 3 is unacceptable. For case (2) with by:
n = 8,nA
na
= 1,n4
=
1,ng
= 3,Ao
2, the best results
2
= 10 l9 GeV,AR
(lowest
An)
are given
= 1.1 x 10s GeV,sinZ8,(AF)
=
0.237, CQ(AF) = 0.117; n = 9, no = 1, n+ = 5, Ao = AR = 1.5x 10” GeV, sin’ &,,(AF) = 0.236, os(A~)
= 0.08. For n = 8 and 9, n+ 2 4 and nd > 6 are unacceptable
The less interesting and n = 8,Ao 0.117(~ 1.5 x 10”
case of doublet
= 1019 GeV,An
GeV,sin*
&,(A,)
AL,~ gives the best value for n+ = na = 3
= 8.8 x 10’s GeV,sin*
= na 2 4 is unacceptable).
respec-
6’,(AF)
= 0.23, and os(A~)
=
For n = 9, one has ng = na = 5,Ao = An =
= 0.241, os(A~)
= 0.08 (higher
values of n+ = na are
unacceptable). An examination
of the above results
of An one can get is around
and Table 1 reveals that
14 TeV. This correspond
the lowest value
to an interesting
lower bound
FERMILAB-Pub-871212-T
-6-
on Mw,,
namely
(Mw,
= igL,nhR):
MwR 2 8 Tel’ We would
like to stress here the fact that
solely, within
the MPP framework,
( 3)
the above lower bound
on Mw,
from sin* 0, and as. The implication
comes
on neutrino
masses is discussed next. Let us reiterate
the results
obtained
In the minimal
model with
ng =
with sin’ &,,(A,)
and
X 8 TeV. These are solely from sin* 0,(A F ) and cys(A~) regardless of how neutrino
In the non-minimal
the bounds
earlier.
MwR X 5 x 10” Gel’, consistent case (n+ = no = 2,n = ~),Mw,
na = 1 and n = 8 generations, os(A~).
obtained
masses come about. In the left-right
symmetric
through
the coupling
Yukawa
couplings
leptons;
and, if AL,n are triplets,
are of the form:
gLd$n,
= r&*rr)
Let us look specifically
Following
masses are obtained
gives < AL >=
and
for leptons only (C is the at the Yukawa couplings
+ $gCraAn$~n)
(v”,
These
for both quarks
Ref. [l] the most general couplings
+ hZ$L$$R + ihs($$CrrAr$L
tion of the potential[‘l
$r$$n($
~~CrZA~$~,~~Crz.An~~
matrix).
of a single lepton family, Ll, = hi$~#n
here, fermion
with the Higgs fields 4 and AL,R (if AL,R are triplets)l’].
Dirac charge-conjugation
Ic
model considered
$=
are given by
+ h.c.. The minimiza( ;R
$=
0
where, in general, K’ < IC (suppression of I?‘, - WR mixing), Vn >> K, and 0 K’ i 1 vL = ~rc*/vR < n(r is the ratio Higgs self-coupling). The charged lepton mass is given by mc- = hld+h2s
while the mass matrix
from (vTiVT)MC
+ h.c.(N
v
i N 1 a = hpL, b = --hpR,c = $(hln+ b > a, c gives the following
the limit formula
7